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| Mirrors > Home > ILE Home > Th. List > elbl2ps | GIF version | ||
| Description: Membership in a ball. (Contributed by NM, 9-Mar-2007.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
| Ref | Expression |
|---|---|
| elbl2ps | β’ (((π· β (PsMetβπ) β§ π β β*) β§ (π β π β§ π΄ β π)) β (π΄ β (π(ballβπ·)π ) β (ππ·π΄) < π )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elblps 14287 | . . . . 5 β’ ((π· β (PsMetβπ) β§ π β π β§ π β β*) β (π΄ β (π(ballβπ·)π ) β (π΄ β π β§ (ππ·π΄) < π ))) | |
| 2 | 1 | 3expa 1205 | . . . 4 β’ (((π· β (PsMetβπ) β§ π β π) β§ π β β*) β (π΄ β (π(ballβπ·)π ) β (π΄ β π β§ (ππ·π΄) < π ))) |
| 3 | 2 | an32s 568 | . . 3 β’ (((π· β (PsMetβπ) β§ π β β*) β§ π β π) β (π΄ β (π(ballβπ·)π ) β (π΄ β π β§ (ππ·π΄) < π ))) |
| 4 | 3 | adantrr 479 | . 2 β’ (((π· β (PsMetβπ) β§ π β β*) β§ (π β π β§ π΄ β π)) β (π΄ β (π(ballβπ·)π ) β (π΄ β π β§ (ππ·π΄) < π ))) |
| 5 | simprr 531 | . . 3 β’ (((π· β (PsMetβπ) β§ π β β*) β§ (π β π β§ π΄ β π)) β π΄ β π) | |
| 6 | 5 | biantrurd 305 | . 2 β’ (((π· β (PsMetβπ) β§ π β β*) β§ (π β π β§ π΄ β π)) β ((ππ·π΄) < π β (π΄ β π β§ (ππ·π΄) < π ))) |
| 7 | 4, 6 | bitr4d 191 | 1 β’ (((π· β (PsMetβπ) β§ π β β*) β§ (π β π β§ π΄ β π)) β (π΄ β (π(ballβπ·)π ) β (ππ·π΄) < π )) |
| Colors of variables: wff set class |
| Syntax hints: β wi 4 β§ wa 104 β wb 105 β wcel 2160 class class class wbr 4018 βcfv 5231 (class class class)co 5891 β*cxr 8009 < clt 8010 PsMetcpsmet 13809 ballcbl 13812 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-map 6668 df-pnf 8012 df-mnf 8013 df-xr 8014 df-psmet 13817 df-bl 13820 |
| This theorem is referenced by: elbl3ps 14291 blcomps 14293 |
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